Department of Mathematics, Faculty of Science, King Mongkut s University of Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand

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1 Abstract and Applied Analysis Volume 2010, Article ID , 22 pages doi: /2010/ Research Article Modified Hybrid Block Iterative Algorithm for Convex Feasibility Problems and Generalized Equilibrium Problems for Uniformly Quasi-φ-Asymptotically Nonexpansive Mappings Siwaporn Saewan and Poom Kumam Department of Mathematics, Faculty of Science, King Mongkut s University of Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand Correspondence should be addressed to Poom Kumam, poom.kum@kmutt.ac.th Received 14 May 2010; Revised 16 June 2010; Accepted 21 June 2010 Academic Editor: W. A. Kirk Copyright q 2010 S. Saewan and P. Kumam. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We introduce a modified block hybrid projection algorithm for solving the convex feasibility problems for an infinite family of closed and uniformly quasi-φ-asymptotically nonexpansive mappings and the set of solutions of the generalized equilibrium problems. We obtain a strong convergence theorem for the sequences generated by this process in a uniformly smooth and strictly convex Banach space with Kadec-Klee property. The results presented in this paper improve and extend some recent results. 1. Introduction and Preliminaries The convex feasibility problem CFP is the problem of computing points laying in the intersection of a finite family of closed convex subsets C j,j 1, 2,...,N, of a Banach space E. This problem appears in various fields of applied mathematics. The theory of optimization 1, Image Reconstruction from projections 2, and Game Theory 3 are some examples. There is a considerable investigation on CFP in the framework of Hilbert spaces which captures applications in various disciplines such as image restoration, computer tomograph, and radiation therapy treatment planning 4. The advantage of a Hilbert space H is that the projection P C onto a closed convex subset C of H is nonexpansive. So projection methods have dominated in the iterative approaches to CFP in Hilbert spaces. In 1993, Kitahara and Takahashi 5 deal with the convex feasibility problem by convex combinations of sunny nonexpansive retractions in a uniformly convex Banach space. It is known that if C

2 2 Abstract and Applied Analysis is a nonempty closed convex subset of a smooth, reflexive, and strictly convex Banach space, then the generalized projection see, Alber 6 or Kamimura and Takahashi 7 from E onto C is relatively nonexpansive, whereas the metric projection from H onto C is not generally nonexpansive. We note that the block iterative method is a method which is often used by many authors to solve the convex feasibility problem CFP see, 8, 9,etc.. In 2008, Plubtieng and Ungchittrakool 10 established strong convergence theorems of block iterative methods for a finite family of relatively nonexpansive mappings in a Banach space by using the hybrid method in mathematical programming. Let C be a nonempty closed convex subset of a real Banach space E with and E being the dual space of E. Letf be a bifunction of C C into R and B : C E a monotone mapping. The generalized equilibrium problem, denoted by GEP, istofindx C such that f ( x, y ) Bx, y x 0, y C. 1.1 The set of solutions for the problem 1.1 is denoted by GEP f, B, thatis GEP ( f, B ) : { x C : f ( x, y ) Bx, y x 0, y C }. 1.2 If B 0, the problem 1.1 reducing into the equilibrium problem for f, denoted by EP f, isto find x C such that f ( x, y ) 0, y C. 1.3 If f 0, the problem 1.1 reducing into the classical variational inequality, denoted by VI B, C, is to find x C such that Bx,y x 0, y C. 1.4 The above formulation 1.3 was shown in 11 to cover monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, optimization problems, variational inequality problems, vector equilibrium problems, and Nash equilibria in noncooperative games. In addition, there are several other problems, for example, the complementarity problem, fixed point problem, and optimization problem, which can also be written in the form of an EP f. In other words, the EP f is a unifying model for several problems arising in physics, engineering, science, optimization, economics, and so forth. In the last two decades, many papers have appeared in the literature on the existence of solutions of EP f ; see, for example 11 and references therein. Some solution methods have been proposed to solve the EP f ; see, for example, and references therein. Consider the functional defined by φ ( x, y ) x 2 2 x, Jy y 2, x, y E, 1.5

3 Abstract and Applied Analysis 3 where J is the duality mapping from E into E. It is well known that if C is a nonempty closed convex subset of a Hilbert space H and P C : H C is the metric projection of H onto C, then P C is nonexpansive. This fact actually characterizes Hilbert spaces and consequently, it is not available in more general Banach spaces. It is obvious from the definition of function φ that ( x y ) 2 φ ( x, y ) ( x y ) 2, x, y E. 1.6 If E is a Hilbert space, then φ x, y x y 2, for all x, y E. On the other hand, the generalized projection Alber 6 Π C : E C is a map that assigns to an arbitrary point x E the minimum point of the functional φ x, y, that is, Π C x x, where x is the solution to the minimization problem φ x, x inf y C φ( y, x ), 1.7 and existence and uniqueness of the operator Π C follow from the properties of the functional φ x, y and strict monotonicity of the mapping J see, for example, 6, 7, Remark 1.1. If E is a reflexive, strictly convex and smooth Banach space, then for x, y E, φ x, y 0ifand only if x y. Itissufficient to show that if φ x, y 0 then x y. From 1.5, we have x y. This implies that x, Jy x 2 Jy 2. From the definition of J, one has Jx Jy. Therefore, we have x y; see 31, 32 for more details. Let C be a closed convex subset of E; a mapping T : C C is said to be nonexpansive if Tx Ty x y, for all x, y C. Apointx C is a fixed point of T provided Tx x. Denote by F T the set of fixed points of T; thatis,f T {x C : Tx x}. Recall that a point p in C is said to be an asymptotic fixed point of T 33 if C contains a sequence {x n } which converges weakly to p such that lim n x n Tx n 0. The set of asymptotic fixed points of T will be denoted by F T. A mapping T from C into itself is said to be relatively nonexpansive if F T F T and φ p, Tx φ p, x for all x C and p F T. The asymptotic behavior of a relatively nonexpansive mapping was studied in T is said to be φ-nonexpansive, if φ Tx,Ty φ x, y for x, y C. T is said to be relatively quasi-nonexpansive if F T / and φ p, Tx φ p, x for all x C and p F T. T is said to be quasi-φ-asymptotically nonexpansive if F T / and there exists a real sequence {k n } 1, with k n 1 such that φ p, T n x k n φ p, x for all n 1x C and p F T. A mapping T is said to be closed if for any sequence {x n } C with x n x and Tx n y, Tx y. Itiseasytoknow that each relatively nonexpansive mapping is closed. The class of quasi-φ-asymptotically nonexpansive mappings contains properly the class of quasi-φ-nonexpansive mappings as a subclass and the class of quasi-φ-nonexpansive mappings contains properly the class of relatively nonexpansive mappings as a subclass, but the converse may be not true see more details A Banach space E is said to be strictly convex if x y /2 < 1 for all x, y E with x y 1andx / y. LetU {x E : x 1} be the unit sphere of E. Then a Banach space E is said to be smooth if the limit lim t 0 x ty x /t exists for each

4 4 Abstract and Applied Analysis x, y U. It is also said to be uniformly smooth if the limit is attained uniformly for x, y U. Let E be a Banach space. The modulus of convexity of E is the function δ : 0, 2 0, 1 defined by δ ε inf{1 x y /2 : x, y E, x y 1, x y ε}. A Banach space E is uniformly convex if and only if δ ε > 0 for all ε 0, 2. Letp be a fixed real number with p 2. A Banach space E is said to be p-uniformly convex if there exists a constant c>0such that δ ε cε p for all ε 0, 2 ; see 42 for more details. Observe that every p-uniform convex is uniformly convex. One should note that no Banach space is p- uniform convex for 1 <p<2. It is well known that a Hilbert space is 2-uniformly convex, uniformly smooth. For each p>1, the generalized duality mapping J p : E 2 E is defined by J p x {x E : x, x x p, x x p 1 } for all x E. In particular, J J 2 is called the normalized duality mapping. If E is a Hilbert space, then J I, where I is the identity mapping. It is also known that if E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E. The following basic properties can be found in Cioranescu 31. i If E is a uniformly smooth Banach space, then J is uniformly continuous on each bounded subset of E. ii If E is a reflexive and strictly convex Banach space, then J 1 is norm-weak - continuous. iii If E is a smooth, strictly convex, and reflexive Banach space, then the normalized duality mapping J : E 2 E is single-valued, one-to-one, and onto. iv A Banach space E is uniformly smooth if and only if E is uniformly convex. v Each uniformly convex Banach space E has the Kadec-Klee property, that is, for any sequence {x n } E, if x n x E and x n x,x n x. In 2005, Matsushita and Takahashi 40 proposed the following hybrid iteration method it is also called the CQ method with generalized projection for relatively nonexpansive mapping T in a Banach space E: x 0 C chosen arbitrarily, y n J 1 α n Jx n 1 α n JTx n, C n { z C : φ ( z, y n ) φ z, xn }, 1.8 Q n {z C : x n z, Jx 0 Jx n 0}, x n 1 Π Cn Q n x 0. They proved that {x n } converges strongly to Π F T x 0, where Π F T is the generalized projection from C onto F T. In 2007, Plubtieng and Ungchittrakool 43 generalized the processes 1.8 to the new general processes of two relatively nonexpansive mappings in a Banach space. Let C be a closed convex subset of a Banach space E and S, T : C C

5 Abstract and Applied Analysis 5 relatively nonexpansive mappings such that F : F S F T /. Define {x n } in the following way: x 0 x C, y n J 1 α n Jx n 1 α n Jz n, z n J 1( β 1 n Jx n β 2 n JTx n β 3 n JSx n ), H n { z C : φ ( z, y n ) φ z, xn }, 1.9 W n {z C : x n z, Jx Jx n 0}, x n 1 P Hn W n x, n 0, 1, 2,..., where {α n }, {β 1 n }, {β 2 n }, and{β 3 n } are sequences in 0, 1 with β 1 n β 2 n β 3 n 1 for all n N {0}. In 2009, Qin et al. 26 introduced a hybrid projection algorithm to find a common element of the set of solutions of an equilibrium problem and the set of common fixed points of two quasi-φ-nonexpansive mappings in the framework of Banach spaces: x 0 x chosen arbitrarily, C 1 C, x 1 Π C1 x 0, y n J 1( α n Jx n β n JTx n γ n JSx n ), 1.10 u n C such that f ( u n,y ) 1 r n y un,ju n Jy n 0, y C, C n 1 { z C n : φ z, u n φ z, x n }, x n 1 Π Cn 1 x 0, where Π Cn 1 is the generalized projection from E onto C n 1. They proved that the sequence {x n } converges strongly to Π F T F S EP f x 0. In the same year, Wattanawitoon and Kumam 44 and Petrot et al. 45 using the idea of Takahashi and Zembayashi 46, 47 and Plubtieng and Ungchittrakool 43 extend the notion from relatively nonexpansive mappings or quasiφ-nonexpansive mappings to two relatively quasi-nonexpansive mappings and also proved some strong convergence theorems to approximate a common fixed point of relatively quasinonexpansive mappings and the set of solutions of an equilibrium problem in the framework of Banach spaces. In 2010, Chang et al. 48 proposed the modified block iterative algorithm for solving the convex feasibility problems for an infinite family of closed and uniformly quasi-φ-asymptotically nonexpansive mapping; they obtain the strong convergence theorems in a Banach space. Recently, many authors considered the iterative methods for finding a common element of the set of solutions to the problem 1.3 and of the set of fixed points of nonexpansive mappings; see, for instance, and the references therein.

6 6 Abstract and Applied Analysis Motivated by Chang et al. 48, Qin et al. 26, 49, Wattanawitoon and Kumam 44, Petrot et al. 45, Zegeye 50, and other recent works, in this paper we introduce a new modified block hybrid projection algorithm for finding a common element of the set of solutions of the generalized equilibrium problems and the set of common fixed points of an infinite family of closed and uniformly quasi-φ-asymptotically nonexpansive mappings which is more general than closed quasi-φ-nonexpansive mappings in a uniformly smooth and strictly convex Banach space E with Kadec-Klee property. The results presented in this paper improve and generalize some well-known results in the literature. 2. Basic Results We also need the following lemmas for the proof of our main results. Lemma 2.1 Kamimura and Takahashi 7. Let E be a uniformly convex and smooth Banach space and let {x n } and {y n } be two sequences of E. Ifφ x n,y n 0 and either {x n } or {y n } is bounded, then x n y n 0. Lemma 2.2 Alber 6. Let C be a nonempty closed convex subset of a smooth Banach space E and x E. Thenx 0 Π C x if and only if x 0 y, Jx Jx 0 0, y C. 2.1 Lemma 2.3 Alber 6. Let E be a reflexive, strictly convex and smooth Banach space, let C be a nonempty closed convex subset of E, and let x E. Then φ ( y, Π C x ) φ Π C x, x φ ( y, x ), y C. 2.2 For solving the equilibrium problem for a bifunction f : C C R, let us assume that f satisfies the following conditions: A1 f x, x 0 for all x C; A2 f is monotone, that is, f x, y f y, x 0 for all x, y C; A3 for each x, y, z C, lim t 0 f ( tz 1 t x, y ) f ( x, y ) ; 2.3 A4 for each x C, y f x, y is convex and lower semicontinuous. Lemma 2.4 Blum and Oettli 11. Let C be a closed convex subset of a smooth, strictly convex and reflexive Banach space E, letf be a bifunction from C C to R satisfying A1 A4, and let r>0 and x E. Then there exists z C such that f ( z, y ) 1 r y z, Jz Jx 0, y C. 2.4

7 Abstract and Applied Analysis 7 Lemma 2.5 Zegeye 50. Let C be a closed convex subset of a uniformly smooth, strictly convex and reflexive Banach space E and let f be a bifunction from C C to R satisfying A1 A4 and let B be a monotone mapping from C into E. For r>0 and x E, define a mapping T r : C C as follows: T r x { z C : f ( z, y ) Bx, y z 1 } y z, Jz Jx 0, y C r 2.5 for all x C. Then the following hold: 1 T r is single-valued; 2 T r is a firmly nonexpansive-type mapping, for all x, y E, Tr x T r y, JT r x JT r y T r x T r y, Jx Jy ; F T r GEP f, B ; 4 GEP f, B is closed and convex. Lemma 2.6 Zegeye 50. (Let C be a closed convex subset of a smooth, strictly convex, and reflexive Banach space E,letf be a bifunction from C C to R satisfying A1 A4, and let B be a monotone mapping from C into E. For r>0, x E, and q F T r, we have that φ ( q, T r x ) φ T r x, x φ ( q, x ). 2.7 Lemma 2.7 Chang et al. 48. Let E be a uniformly convex Banach space, r>0apositive number, and B r 0 a closed ball of E. Then, for any given sequence {x i } B r 0 and for any given sequence {λ i } of positive number with n 1 λ n 1, there exists a continuous, strictly increasing, and convex function g : 0, 2r 0, with g 0 0 such that, for any positive integer i, j with i<j, 2 λ n x n λ n x n 2 λ i λ j g ( ) xi x j. n 1 n Lemma 2.8 Chang et al. 48. Let E be a real uniformly smooth and strictly convex Banach space, and C a nonempty closed convex subset of E. LetT : C C be a closed and quasi-φ-asymptotically nonexpansive mapping with a sequence {k n } 1,, k n 1. ThenF T is a closed convex subset of C. Definition 2.9 Chang et al Let {S i } : C C be a sequence of mapping. {S i } is said to be a family of uniformly quasi-φ-asymptotically nonexpansive mappings, if n 1 F S n /, and there exists a sequence {k n } 1, with k n 1 such that for each i 1, φ ( p, S n i x) k n φ ( p, x ), p n 1 F S n, x C, n

8 8 Abstract and Applied Analysis 2 A mapping S : C C is said to be uniformly L-Lipschitz continuous if there exists a constant L>0 such that S n x S n y L x y, x, y C Main Results In this section, we prove the new convergence theorems for finding the set of solutions of a general equilibrium problems and the common fixed point set of a family of closed and uniformly quasi-φ-asymptotically nonexpansive mappings in a uniformly smooth and strictly convex Banach space E with Kadec-Klee property. Theorem 3.1. Let C be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space E with Kadec-Klee property. Let f be a bifunction from C C to R satisfying A1 A4. LetB be a continuous monotone mapping of C into E.Let{S i } : C C be an infinite family of closed uniformly L i -Lipschitz continuous and uniformly quasi-φ-asymptotically nonexpansive mappings with a sequence {k n } 1,, k n 1 such that F : F S i GEP f, B is a nonempty and bounded subset in C. For an initial point x 0 E with x 1 Π C1 x 0 and C 1 C, we define the sequence {x n } as follows: y n J 1( β n Jx n ( ) ) 1 β n Jzn, ) z n J (α 1 n,0 Jx n α n,i JS n i x n, u n C such that f ( u n,y ) By n,y u n 1 r n y u n,ju n Jy n 0, y C, 3.1 C n 1 { z C n : φ z, u n φ z, x n θ n }, x n 1 Π Cn 1 x 0, n 0, where J is the duality mapping on E, θ n sup q F k n 1 φ q, x n, {α n,i }, {β n } are sequences in 0, 1 and {r n } a, for some a>0. If i 0 α n,i 1 for all n 0 and lim inf n α n,0 α n,i > 0 for all i 1, then{x n } converges strongly to p F, wherep Π F x 0. Proof. We first show that C n 1 is closed and convex for each n 0. Clearly C 1 C is closed and convex. Suppose that C n is closed and convex for each n N. Since for any z C n,we know φ z, u n φ z, x n θ n 2 z, Jx n Ju n x n 2 u n 2 θ n. 3.2 So, C n 1 is closed and convex. Therefore, Π F x 0 and Π Cn x 0 are well defined. Next, we show that F C n for all n 0. Indeed, put u n T rn y n for all n 0. It is clear that F 1 C 1 C.

9 Abstract and Applied Analysis 9 Suppose F C n for n N, by the convexity of 2, property of φ, Lemma 2.7, and uniformly quasi-φ-asymptotically nonexpansive of S n for each q F C n, we observed that φ ( ) ( ) q, u n φ q, Trn y n φ ( ) q, y n φ q, J 1( β n Jx n ( ) ) 1 β n Jzn q 2 2 q, βn Jx n ( ) 1 β n Jzn βn Jx n 1 β n Jz n q 2 2βn q, Jxn 2 ( 1 βn ) q, Jzn βn x n 2 ( 1 β n ) zn 2 β n φ ( q, x n ) ( 1 βn ) φ ( q, zn ) and φ ( ( )) ) q, z n φ q, J (α 1 n,0 Jx n α n,i JS n i x n q 2 2 q, α n,0 Jx n α n,i JS n i x n α n,0jx n α n,i JS n i x n q 2 2αn,0 q, Jxn 2 α n,i q, JS n i x n α n,0 Jx n α n,i JS n i x n 2 2 q 2 2αn,0 q, Jx n 2 α n,i q, JS n i x n αn,0 Jx n 2 α n,i JS n α n,0 α n,j gjx n JS n j x n i x 2 n 3.4 q 2 2α n,0 q, Jxn αn,0 Jx n 2 2 α n,i q, JS n i x n α n,i JS n i x n 2 α n,0 α n,j gjx n JS n j x n α n,0 φ ( ) q, x n α n,i φ ( q, S n i x n) αn,0 α n,j gjx n JS n j x n α n,0 k n φ ( ) q, x n α n,i k n φ ( ) q, x n αn,0 α n,j gjx n JS n j x n k n φ ( q, x n ).

10 10 Abstract and Applied Analysis Substituting 3.4 into 3.3, weget φ ( q, u n ) βn φ ( q, x n ) ( 1 βn ) φ ( q, zn ) β n φ ( q, x n ) ( 1 βn ) kn φ ( q, x n ) β n φ ( ) ( ) [ q, x n 1 βn φ ( ) q, x n sup k n 1 φ ( ) ] q, x n q F φ ( ) ( ) q, x n 1 βn sup k n 1 φ ( ) q, x n q F 3.5 φ ( q, x n ) θn. This show that q C n 1 implies that F C n 1 and hence, F C n for all n 0. Since F is nonempty, C n is a nonempty closed convex subset of E, and hence Π Cn exist for all n 0. This implies that the sequence {x n } is well defined. From definition of C n 1 that x n Π Cn x 0 and x n 1 Π Cn 1 x 0, we have φ x n,x 0 φ x n 1,x 0, n By Lemma 2.3, we also have φ x n,x 0 φ Π Cn x 0,x 0 φ ( ) ( ) q, x 0 φ q, xn 3.7 φ ( q, x 0 ), q F Cn, n 0. From 3.6 and 3.7, then {φ x n,x 0 } are nondecreasing and bounded. So, we obtain that lim n φ x n,x 0 exists. In particular, by 1.6, the sequence { x n x 0 2 } is bounded. This implies that {x n } is also bounded. Denote K sup{ x n } <. n Moreover, by the definition of {θ n } and 3.8, it follows that θ n 0, n. 3.9 Next, we show that {x n } is a Cauchy sequence in C. Since x m Π Cm x 0 C m C n,for m>n,bylemma 2.3, we have φ x m,x n φ x m, Π Cn x 0 φ x m,x 0 φ Π Cn x 0,x φ x m,x 0 φ x n,x 0.

11 Abstract and Applied Analysis 11 Since lim n φ x n,x 0 exists and we take m, n, then, we get φ x m,x n 0. From Lemma 2.1, we have lim n x m x n 0. Thus {x n } is a Cauchy sequence and by the completeness of E, and there exists a point p C such that x n p as n. Now, we claim that Ju n Jx n 0, as n. By definition of Π Cn x 0, one has φ x n 1,x n φ x n 1, Π Cn x 0 φ x n 1,x 0 φ Π Cn x 0,x φ x n 1,x 0 φ x n,x 0. Since lim n φ x n,x 0 exists, we have lim n φ x n 1,x n By Lemma 2.1, weobtainthat lim n x n 1 x n Since J is uniformly norm-to-norm continuous on bounded subsets of E, weget lim Jx n 1 Jx n 0. n 3.14 From x n 1 Π Cn 1 x 0 C n 1 C n and the definition of C n 1, we have φ x n 1,u n φ x n 1,x n θ n By 3.9 and 3.12, we also have lim n φ x n 1,u n Applying Lemma 2.1, weobtain lim x n 1 u n 0. n 3.17 Since u n x n u n x n 1 x n 1 x n, we get lim n u n x n Since J is uniformly norm-to-norm continuous on bounded subsets of E, we have lim n Ju n Jx n

12 12 Abstract and Applied Analysis Next, we will show that x n p F : GEP f, B F S i. i First, we show that x n p GEP f, B. It follows from 3.3 and 3.4 we observe that φ p, y n φ p, x n θ n. By Lemma 2.6 and u n T rn y n,weobtain φ ( u n,y n ) φ ( Trn y n,y n ) φ ( p, y n ) φ ( p, Trn y n ) φ ( p, x n ) φ ( p, Trn y n ) θn φ ( p, x n ) φ ( p, un ) θn p 2 2 p, Jx n x n 2 ( p 2 2 p, Ju n u n 2) θ n 3.20 x n 2 u n 2 2 p, Jx n Ju n θ n x n u n x n u n 2 p Jx n Ju n θ n. By Lemma 2.1, 3.9, 3.18,and 3.19, we have lim u n y n 0. n 3.21 Again since J is uniformly norm-to-norm continuous, we also have lim Jun Jy n 0. n 3.22 From A2, wenotethat Byn,y u n 1 r n y un,ju n Jy n f ( un,y ) f ( y, u n ), y C, 3.23 and hence Byn,y u n y u n, Ju n Jy n f ( ) y, u n, y C r n For t with 0 <t<1andy C, let y t ty 1 t p. Then y t C and hence 0 By n,y t u n y t u n, Ju n Jy n f ( ) y t,u n, y C r n

13 Abstract and Applied Analysis 13 It follows that By t,y t u n By t,y t u n By n,y t u n y t u n, Ju n Jy n r n By t,y t u n Bun,y t u n Bun,y t u n By n,y t u n y t u n, Ju n Jy n f ( ) y t,u n, r n f ( y t,u n ), yt C By t Bu n,y t u n Bun By n,y t u n y t u n, Ju n Jy n f ( y t,u n ), yt C. r n yt C 3.26 Since x n p as n from 3.18 and 3.21, we can get u n p and y n p as n. Furthermore, it follows from the continuity of B that Bu n By n 0asn.Fromr n > 0 and 3.22, we have Ju n Jy n /r n 0asn. Since B is monotone, we know that By t Bu n,y t u n 0. Thus, it follows from A4 that f ( y t,p ) lim inf f( ) y t,u n n lim n By t,y t u n 3.27 By t,y t p. From the conditions A1 and A4, we have 0 f ( y t,y t ) tf ( y t,y ) 1 t f ( y t,p ) tf ( y t,y ) 1 t By t,y t p 3.28 tf ( y t,y ) 1 t t By t,y p, and hence 0 f ( y t,y ) 1 t By t,y p Letting t 0, we get 0 f ( p, y ) Bp, y p, y C This implies that p GEP f, B.

14 14 Abstract and Applied Analysis ii We show that x n p F S i. From definition of C n 1 and since x n 1 Π Cn 1 x 0 C n 1, we have φ x n 1,z n φ x n 1,x n θ n Form Lemma 2.1 and 3.9, weobtainthat lim n x n 1 z n Since J is uniformly norm-to-norm continuous, we also have lim n Jx n 1 Jz n From 3.1, we compute ( Jx n 1 Jz n Jx n 1 α n,0 Jx n α n,i JS n i n) x α n,0jx n 1 α n,0 Jx n α n,i Jx n 1 α n,i JS n i x n α ( n,0 Jx n 1 Jx n α n,i Jxn 1 JS n i x ) n ( α n,i Jxn 1 JS n i x n) αn,0 Jx n Jx n 1 α Jxn 1 n,i JS n i x n αn,0 Jx n Jx n 1, 3.34 and hence Jxn 1 JS n i x 1 n α Jx n 1 Jz n α n,0 Jx n Jx n n,i From 3.14, 3.33, and lim inf n α n,i > 0, we get lim Jxn 1 JS n i x n 0. n 3.36 Since J 1 is uniformly norm-to-norm continuous on bounded sets, we obtain that lim x n 1 S n i x n 0. n 3.37

15 Abstract and Applied Analysis 15 By the triangle inequality, xn S n i x n xn x n 1 x n 1 S n i x n x n x n 1 xn 1 S n i x n Hence from 3.13 and 3.37, we have lim xn S n i x n 0. n 3.39 Since J is uniformly continuous on any bounded subset of E, we obtain lim Jx n JS n i x n 0, i n Since x n p and J is uniformly continuous, it yields Jx n Jp.Thusfrom 3.40, weget JS n i x n Jp, i Since J 1 : E E is norm-weak -continuous, we have S n i x n p, i On the other hand, for each i 1, we observe that S n i x n p J ( S n i x n) Jp J ( S n i x n) Jp In view of 3.41,weobtain S n i x n p for each i 1. Since E has the Kadec-Klee property, we get S n i x n p, for each i By the assumption that for each i 1,S i is uniformly L i -Lipschitz continuous, we have S n 1 i x n S n i x n S n 1 i x n S n 1 S i x n 1 n 1 i x n 1 x n 1 xn 1 x n xn S n i x n L i 1 x n 1 x n S n 1 i x n 1 x n 1 xn S n i x n From 3.13 and 3.39, it yields that S n 1 i x n S n i x n 0. From S n i x n p,wegets n 1 i x n p, that is S i S n i x n p. In view of closeness of S i, we have S i p p, for all i 1. This implies that p F S i.

16 16 Abstract and Applied Analysis Finally, we show that x n p Π F x 0.Letu Π F x 0.Fromx n Π Cn x 0 and u F C n, we have φ x n,x 0 φ u, x 0, n This implies that φ ( p, x 0 ) lim n φ x n,x 0 φ u, x By definition of p Π F x 0, we have p u. Therefore, x n proof. p Π F x 0. This completes the If S i S for each i N, then Theorem 3.1 is reduced to the following corollary. Corollary 3.2. Let C be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space E with Kadec-Klee property. Let f be a bifunction from C C to R satisfying A1 A4. LetB be a continuous monotone mapping of C into E.LetS : C C be a closed uniformly L-Lipschitz continuous and quasi-φ-asymptotically nonexpansive mappings with a sequence {k n } 1,, k n 1, such that F : F S GEP f, B is a nonempty and bounded subset in C. For an initial point x 0 E with x 1 Π C1 x 0 and C 1 C, we define the sequence {x n } as follows: y n J 1( β n Jx n ( 1 β n ) Jzn ), z n J 1 α n Jx n 1 α n JS n x n, u n C such that f ( u n,y ) By n,y u n 1 r n y u n,ju n Jy n 0, y C, 3.48 C n 1 { z C n : φ z, u n φ z, x n θ n }, x n 1 Π Cn 1 x 0, n 0, where J is the duality mapping on E, θ n sup q F k n 1 φ q, x n, {α n }, {β n } are sequences in 0, 1, and {r n } a, for some a>0. Iflim inf n α n 1 α n > 0, then{x n } converges strongly to p F, where p Π F x 0. For a special case that i 1, 2, we can obtain the following results on a pair of quasi-φasymptotically nonexpansive mappings immediately from Theorem 3.1. Corollary 3.3. Let C be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space E with Kadec-Klee property. Let f be a bifunction from C C to R satisfying A1 A4. LetB be a continuous monotone mapping of C into E.LetS, T : C C be two closed quasi-φ-asymptotically nonexpansive mappings and uniformly L S,L T -Lipschitz continuous, respectively with a sequence {k n } 1,, k n 1 such that F : F S F T GEP f, B is a

17 Abstract and Applied Analysis 17 nonempty and bounded subset in C. For an initial point x 0 E with x 1 Π C1 x 0 and C 1 C, we define the sequence {x n } as follows: y n J 1( β n Jx n ( 1 β n ) Jzn ), z n J 1( α n Jx n β n JS n x n γ n JT n x n ), u n C such that f ( u n,y ) By n,y u n 1 r n y un,ju n Jy n 0, y C, 3.49 C n 1 { z C n : φ z, u n φ z, x n θ n }, x n 1 Π Cn 1 x 0, n 0, where J is the duality mapping on E, θ n sup q F k n 1 φ q, x n, {α n,i }, {β n } are sequences in 0, 1, and {r n } a, for some a>0. Ifα n β n γ n 1 for all n 0 and lim inf n α n β n > 0 and lim inf n α n γ n > 0, then{x n } converges strongly to p F,wherep Π F x 0. Remark 3.4. Corollary 3.3 improves and extends 44, Theorem 3.1 in the following senses: i for the mappings, we extend the mappings from two closed relatively quasinonexpansive mappings to an infinite family of closed and uniformly quasi-φasymptotically mappings, ii from a solution of the classical equilibrium problem to the generalized equilibrium problem, iii for the framework of spaces, we extend the space from a uniformly smooth and uniformly convex Banach space to a uniformly smooth and strictly convex Banach space with the Kadec-Klee property. Corollary 3.5. Let C be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space E with Kadec-Klee property. Let f be a bifunction from C C to R satisfying A1 A4. LetB be a continuous monotone mapping of C into E.Let{S i } : C C be an infinite family of closed quasi-φ-nonexpansive mappings such that F : F S i GEP f, B /. For an initial point x 0 E with x 1 Π C1 x 0 and C 1 C, we define the sequence {x n } as follows: y n J 1( β n Jx n ( 1 β n ) Jzn ), z n J 1 (α n,0 Jx n α n,i JS i x n ), u n C such that f ( u n,y ) By n,y u n 1 r n y un,ju n Jy n, y C, 3.50 C n 1 { z C n : φ z, u n φ z, x n }, x n 1 Π Cn 1 x 0, n 0, where J is the duality mapping on E, {α n,i }, {β n } are sequences in 0, 1, and {r n } a, for some a>0. If i 0 α n,i 1 for all n 0 and lim inf n α n,0 α n,i > 0 for all i 1, then{x n } converges strongly to p F,wherep Π F x 0.

18 18 Abstract and Applied Analysis Proof. Since {S i } : C C is an infinite family of closed quasi-φ-nonexpansive mappings, it is an infinite family of closed and uniformly quasi-φ-asymptotically nonexpansive mappings with sequence k n 1. Hence the conditions appearing in Theorem 3.1 F is a bounded subset in C and for each i 1,S i is uniformly L i -Lipschitz continuous are of no use here. By virtue of the closeness of mapping S i for each i 1, it yields that p F S i for each i 1, that is, p F S i. Therefore, all conditions in Theorem 3.1 are satisfied. The conclusion of Corollary 3.5 is obtained from Theorem 3.1 immediately. Corollary 3.6. Let C be a nonempty closed and convex subset of a uniformly smooth and strictly convex Banach space E with Kadec-Klee property. Let f be a bifunction from C C to R satisfying A1 A4.LetB be a continuous monotone mapping of C into E.Let{S i } : C C be an infinite family of closed and uniformly quasi-φ-asymptotically nonexpansive mappings with a sequence {k n } 1,, k n 1 and uniformly L i -Lipschitz continuous such that F : F S i GEP f, B is a nonempty and bounded subset in C. For an initial point x 0 E with x 1 Π C1 x 0 and C 1 C, we define the sequence {x n } as follows: y n J 1 (α n,0 Jx n ) α n,i JS n i x n, u n C such that f ( u n,y ) By n,y u n 1 r n y un,ju n Jy n 0, y C, 3.51 C n 1 { z C n : φ z, u n φ z, x n θ n }, x n 1 Π Cn 1 x 0, n 0, where J is the duality mapping on E, θ n sup q F k n 1 φ q, x n, {α n,i } is a sequence in 0, 1, and {r n } a, for some a>0. If i 0 α n,i 1 for all n 0 and lim inf n α n,0 α n,i > 0 for all i 1, then {x n } converges strongly to p F,wherep Π F x 0. Proof. Setting β n 0inTheorem 3.1, then, we get that y n z n. Thus, the method of proof of Theorem 3.1 we obtain Corollary 3.6 immediately. Remark 3.7. Theorem 3.1, Corollary 3.3, and Corollary 3.5, improve and extend the corresponding results in Qin et al. 49 and Zegeye 50 in the following senses: i from a solution of the classical equilibrium problem to the generalized equilibrium problem with an infinite family of quasi-φ-asymptotically mappings, ii for the mappings, we extend the mappings from nonexpansive mappings, relatively quasi-nonexpansive mappings or quasi-φ-nonexpansive mappings and a finite family of closed relatively quasi-nonexpansive mappings to an infinite family of quasi-φ-asymptotically nonexpansive mappings, iii for the framework of spaces, we extend the space from a uniformly smooth and uniformly convex Banach space to a uniformly smooth and strictly convex Banach space with the Kadec-Klee property.

19 Abstract and Applied Analysis Applications If E H, a Hilbert space, then H is uniformly smooth and strictly convex Banach space E with Kadec-Klee property and closed relatively quasi-nonexpansive mappings reducing to closed quasi-nonexpansive mappings. Moreover, J I, identity operator on H and Π C P C, projection mapping from H into C. Thus, the following corollaries hold. Theorem 4.1. Let C be a nonempty closed and convex subset of a Hilbert space H. Letf be a bifunction from C C to R satisfying A1 A4. LetB be a continuous monotone mapping of C into H. Let{S i } : C C be an infinite family of closed and uniformly quasi-φ-asymptotically nonexpansive mappings with a sequence {k n } 1,, k n 1 and uniformly L i -Lipschitz continuous such that F : F S i GEP f, B is a nonempty and bounded subset in C. For an initial point x 0 H with x 1 P C1 x 0 and C 1 C, we define the sequence {x n } as follows: y n β n x n ( 1 β n ) zn, z n α n,0 x n α n,i S n i x n, u n C such that f u n, y By n,y u n 1 r n y un,u n y n 0, y C, 4.1 C n 1 {z C n : z u n α n z x n 1 α n z z n z x n θ n }, x n 1 P Cn 1 x 0, n 0, where θ n sup q F k n 1 q x n, {α n,i }, {β n } are sequences in 0, 1, and {r n } a, for some a>0. If i 0 α n,i 1 for all n 0 and lim inf n α n,0 α n,i > 0 for all i 1, then{x n } converges strongly to p F,wherep Π F x 0. Corollary 4.2. Let C be a nonempty closed and convex subset of a Hilbert space H. Letf be a bifunction from C C to R satisfying A1 A4. LetB be a continuous monotone mapping of C into H. Let{S i } : C C be an infinite family of closed and quasi-φ-nonexpansive mappings with a sequence {k n } 1,, k n 1 and uniformly L i -Lipschitz continuous such that F : F S i GEP f, B is a nonempty and bounded subset in C. For an initial point x 0 H with x 1 P C1 x 0 and C 1 C, we define the sequence {x n } as follows: y n β n x n ( 1 β n ) zn, z n α n,0 x n α n,i S n i x n, u n C such that f ( u n,y ) By n,y u n 1 r n y un,u n y n 0, y C, 4.2 C n 1 {z C n : z u n α n z x n 1 α n z z n z x n }, x n 1 P Cn 1 x 0, n 0, where {α n,i }, {β n } are sequences in 0, 1 and {r n } a, for some a>0.if i 0 α n,i 1 for all n 0 and lim inf n α n,0 α n,i > 0 for all i 1, then{x n } converges strongly to p F,wherep Π F x 0.

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